Hamiltonian systems in the plane involving vanishing potentials and exponential growth

In this work, we guarantee the existence of positive solution for the following class of Hamiltonian systems: [Formula: see text] where the potential [Formula: see text] can decay to zero at infinity as [Formula: see text], with [Formula: see text], and [Formula: see text] decay to zero at infinity as [Formula: see text] with [Formula: see text]. The nonlinear terms [Formula: see text] and [Formula: see text] have exponential subcritical or critical growth. First, we present an alternative proof of a weighted Trudinger–Moser type inequality. Then, we combine this inequality with the Galerkin approximation method and a linking theorem to establish our existence result.